As we will see later, the charged lepton Yukawa is already diagonalized in the flavor basis. As pointed out in Ref. [1]}, if we further introduce three right-handed neutrinos with flavor charges \((1, -1, 0)\) , and if the VEVs of the \(B-L\)  Higgs are of similar magnitude, then light neutrino masses and the large mixing angles can be realized by the seesaw mechanism [2]}, [3]}, [4]}, [5]}. In the following we will focus on the correlation of the axion and heavy Higgs bosons.
